Department of Mathematics & Statistics Prof. M. Rahman

MAC 2233, REVIEW FINAL(Spring 2007) SE1. In the state of Florida the sales tax T on the amount of taxable goods is 6% of the value of the goods purchased (x), where both T and x are measured in dollars.

(i) Express T as a function of x. and (ii) Find T(400) and T(80.5).

SE2. If f (x)  2  3x  2x 2 and g(x)  2x 1, find the following : f (2  h)  f (2) f (2  h)  f (2) (a) f ° g(2) (b) and lim h h0 h SE3.Find the following limits: x 2  7x 10 1 h 1 (i) (ii) lim 2 lim x5 x 11x  30 h0 h x 2  4 x 2  6x  8 3x  2 (iii) (iv) (v) lim lim 2 lim x2 x  2 x4 x 16 x x  5 (vi). F(x)  x  3 for  2  x  0  1 2x for 0  x  2 Find lim F(x) x0 SE4. f (x)  2x 1 when  2  x  0 Find f (2), f (0), f (1), and f (0.5).  x 2 when 0  x  2  6  x when 2  x  4 find lim f (x) and lim f (x) x0 x2

SE5. USING RULES OF DIFFERENTIATION , find f (x) if 1 f (x)  f (x)  x 4  50x 2  20x  496 x

7x 2  8x 10 f (x)  33 x  2 x f (x)  5 SE6. USING RULES DO THE FOLLOWING:

a.)If G(x)  (3x 2  8x  5)(81 20x 10x 3 ), then find G(x) without multiplying the factors. Do not simplify your answer. b.)Find G(2) and simplify your answer for this. x 2  4 c.) F(x)  , find F(x) and simplify your answer. x  2 x  2x 2  3x3 d.) f (x)  x 2 SE7. Applications problems: Section # 64(Profit from sale of pagers);Section 2.3 # 18, 40, 74, 76; Section 3.1 # 58; Section, 3.2 # 55

SP8.(a) Find the derivative:

1 Department of Mathematics & Statistics Prof. M. Rahman

d d  1  d  2x 1  d (i) ( 3x  2), (ii)  , (iii)  , (iv) (3x 1)4 (x 2  x 1)3    2    dx dx  3x  2  dx  x 1  dx (b) Section 3.3--# 4, 6,7, 30 © Section 3.4- Ex 6, 8, 10, Example 5, Example 6 SP9.(a) Find the equation of the tangent line at x  1 on y  5  4x . (b) For y  f (x)  (5x 2  6)20 calculate f (x) . x (c) Explain why f (x)  is always decreasing. 4x  3 6 (d) Explain why the graph of y  5x 2  is always concave up. x 2

SP10. For y  f (x)  8  6x 2  x3 , answer the following: (a) When is the function (i) increasing? (ii) decreasing? (b) When is the graph (i) concave up? (ii) concave down? (c) Find the point of inflection. (d) Find relative max and relative min. (e) Draw the graph for  2  x  6.

SP11. Find the intervals where the graphs of the following functions are concave up and the intervals where concave down: 1 (i) f (x)  6x  x 2 (ii) g(x)  x 2  x 2

1 (iii) F(x)  x 4  x 3  5x 10 4

SP12. Find relative max and relative min for y  f (x)  x3  3x 2 10 .

SP13 (a) A company manufactures x televisions per month. The cost function C(x) 100,000 100x and x the unit sale price  p  500  (0  x  10,000 ). 20 How many televisions must be manufactured every month for maximum profit? And what is the maximum profit?

(b) Section 4.3—12, 22, 41, 42; Section 4.4(Optimization)—Exercise 48; Section 4.5(Optimization)- Example 1(page 316), Exercise- # 6,8,15

SP 14.(a) Differentiate the following: 2 (i) f (x)  e x 5x6 (ii) g(x)  ln(x 2e5x ) (iii) R(x)  (5x 1)3 (2x 1)5

5 (b) Find the equation of the tangent line at x = -2 on y  f (x)  . 4  3x

2 Department of Mathematics & Statistics Prof. M. Rahman

f (x)  f (2) (c) Find lim where f (x)  5x 2  8x 108. x2 x  2

SP15. (a) Find intervals where (i) f (x)  50 15x  6x 2  x3 is increasin g and is decreasin g (ii) the graph of g(x)  x 4  6x 2 100 is concave up and is concave down.

(b) Find f (x) if f (x)  (x5 1)8 . 16 (c) Find relative max and relative min for y  f (x)  100  x  . x SP16. The cost of manufacturing x televisions is, C(x)  5000  600x  0.01x 2 . If each set is sold at $ 1000 , how many television sets should be manufactured for maximum profit?

SP17.(a) Find the derivatives of exponential and logarithmic functions. (b) Graph the exponential and logarithmic functions using pencil and paper. © Compound interest problem( do related problem that I did in class)

18 1 SP18. Evaluate  dx 4 4x  9

SP19.(a) Integrate 2 5 1 (i) (  3 x  )dx (ii) dx (iii) e5x6 dx  x x 2  3x  5 

(b) C(x)  6x 2 12x 100 where C(x) is the cost of producing x items. If the cost of producing 10 items is $5000, the what will be the cost of producing 20 items?